A function g is defined by \(\mathrm{g(x) = a(x-h)^2 + k}\), where a, h, and k are constants. In the...

1. TRANSLATE the vertex information

• Given information:
  • Vertex is at (4, -10)
  • In vertex form \(\mathrm{g(x) = a(x-h)^2 + k}\), the vertex is \(\mathrm{(h, k)}\)
• What this tells us: \(\mathrm{h = 4}\) and \(\mathrm{k = -10}\)

2. TRANSLATE the point information

• Given: The graph passes through (6, -2)
• What this means: When \(\mathrm{x = 6}\), the function value \(\mathrm{g(6) = -2}\)

3. INFER the solution strategy

• We have \(\mathrm{h = 4}\) and \(\mathrm{k = -10}\), but we need to find 'a'
• Since we know a point on the graph, we can substitute it into our equation to solve for 'a'
• Our equation becomes: \(\mathrm{g(x) = a(x-4)^2 - 10}\)

4. SIMPLIFY to find the value of 'a'

• Substitute the point (6, -2): \(\mathrm{-2 = a(6-4)^2 - 10}\)
• Simplify inside parentheses: \(\mathrm{-2 = a(2)^2 - 10}\)
• Evaluate the square: \(\mathrm{-2 = 4a - 10}\)
• Add 10 to both sides: \(\mathrm{8 = 4a}\)
• Divide by 4: \(\mathrm{a = 2}\)

5. INFER the final calculation

• The question asks for \(\mathrm{a + h}\)
• We found: \(\mathrm{a = 2}\), \(\mathrm{h = 4}\)
• Therefore: \(\mathrm{a + h = 2 + 4 = 6}\)

Answer: C) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which values correspond to h and k from the vertex coordinates.

They might think the vertex (4, -10) means \(\mathrm{h = -10}\) and \(\mathrm{k = 4}\), mixing up the order. This leads to setting up the wrong equation from the start, making all subsequent work incorrect. This typically leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when solving \(\mathrm{-2 = 4a - 10}\).

Common mistakes include forgetting to add 10 to both sides, or incorrectly calculating 8 ÷ 4. If they get \(\mathrm{a = 1}\) instead of \(\mathrm{a = 2}\), they would calculate \(\mathrm{a + h = 1 + 4 = 5}\), but since 5 isn't an answer choice, this leads to confusion and random selection.

The Bottom Line:

This problem tests whether students truly understand the vertex form structure and can accurately translate word descriptions into mathematical relationships. Success requires precise attention to coordinate ordering and careful algebraic manipulation.